I think that's a poor characterization of the purpose of mathematical logic. The purpose of mathematical logic is not to capture the human mind, it's to overcome the limitations of the human mind.

Math is like sailing a ship during the age of sail. In the middle of the ocean, there is no point of reference. You have a vague sense of direction, but it's easy to get lost. You need some tools to navigate. Math is the ocean, your mind is the ship and mathematical logic is the navigation instruments.